| Abstract: |
| Recent simulations of ecosystem models (of reaction-diffusion type) have shown that a Turing bifurcation preceding a saddle-node bifurcation may enable an ecosystem to evade tipping: under slowly changing circumstances, an ecosystem may form patterns instead of collapsing into a less desirable state. Thus, the formation of patterns may increase the resilience of the ecosystem. In this talk we present the system of coupled modulation equations that governs the dynamics of small amplitude patterns near a co-dimension 2 Turing-fold bifurcation. First we show that there is a critical coefficient -- that can be determined explicitly and that corresponds to the Landau coefficient in the classical (co-dimension 1) Ginzburg-Landau setting -- that decides whether the system tips or evades tipping by forming (stable) patterns. Moreover, we show that the boundary of the region in (wavenumber,parameter)-space of stable periodic solutions -- i.e. the Busse balloon -- is much richer than that in the classical Ginzburg-Landau case. Instead of only a sideband instability, patterns can also be destabilized by several Turing- and Turing-Hopf-type mechanisms. The associated bifurcations may first lead to the formation of stable stationary quasi-periodic patterns and subsequently to time-periodic and eventually irregular dynamics. |
|